Dilatation monotone risk measures are law invariant

Abstract

Coherent risk measures, Convex risk measures, Dilatation monotonicity, Factor monotonicity, Fatou property, Law invariance, Second-order stochastic dominance, G32, 91B16, 91B30,

Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders

“…On the other hand, we know from Cherny and Grigoriev (2007) that a law-invariant and coherent risk measure necessarily respects stop-loss order. Consequently, Axioms (a2), (a3), (A4) and Axiom SL are equivalent to Axioms (a2), (a3), (a4) and Axiom ST.…”

Shortfall deviation risk: an alternative for risk measurement

“…If the measure satisfies the Dilated Monotonicity axiom, it is guaranteed that if ≼ , then ( ) ≥ ( ), ∀ , ∈ . SDR satisfies this axiom because, as demonstrated by Cherny and Grigoriev (2007), every convex risk measure satisfying the Law Invariance axiom, which is defined in atomless probability space, also satisfies the Dilated Monotonicity axiom. Based on these properties, SDR becomes a very interesting measure for decision-making applications because it reflects preference relations.…”

Optimal and robust contracts for a risk-constrained principal

“…And in any case, in the story we are telling, there is no scope for cleverness or innovation on the part of the trader; whatever he may believe about his god-like insights, he is working in a complete market, and the only thing 1 While initially appealing, there are reasons why this may not be desirable; a trader would not necessarily be indifferent between two contingent claims with the same law, one of which was the market portfolio, the other being negatively correlated with the market portfolio. In a similar spirit, Cherny & Grigoriev [4] point out that law invariance would not be a natural assumption when an agent has an existing position to be offset against the proposed contingent claim.…”

“…We shall suppose that there is a complete market with a single risky asset 4 , where the dynamics of the wealth process w take the familiar form…”

“…We now modify the problem; the objective remains to maximise EU (w T ) over admissible portfolio processes θ, but now subject to further constraints which may be expressed solely in 4 There would be no difficulty in extending the formulation to multiple risky assets, with suitably bounded processes for volatility and growth rate; the key assumption is that the market is complete. We take the unvariate constant coefficient case only to allow us to do explicit calculations for a number of examples.…”